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Approximation of functions of finite variation by superpositions of a sigmoidal function. (English) Zbl 1067.41013

Summary: The aim of this note is to generalize a result of A. R. Barron [IEEE Trans. Inf. Theory 39, No. 3, 930–945 (1993)] concerning the approximation of functions, which can be expressed in terms of the Fourier transform, by superpositions of a fixed sigmoidal function. In particular, we consider functions of the type \(h(x)=\int_{\mathbb{R}^d} f(\langle t,x\rangle)d\mu (t)\), where \(\mu\) is a finite Radon measure on \(\mathbb{R}^d\) and \(f:\mathbb{R}\to\mathbb{C}\) is a continuous function with bounded variation in \(\mathbb{R}\). We show (Theorem 2.6) that these functions can be approximated in \(L_2\)-norm by elements of the set \(G_n= \{\sum^n_{i=0} c_ig(\langle a_i,x\rangle+b_i):a_i\in\mathbb{R}^d,b_kc_i\in \mathbb{R}\}\), where \(g\) is a fixed sigmoidal function, with the error estimated by \(C/n^{1/2}\), where \(C\) is a positive constant depending only on \(f\). The same result holds true (Theorem 2.9) for \(f:\mathbb{R}\to \mathbb{C}\) satisfying the Lipschitz condition under an additional assumption that \(\int_{\mathbb{R}^d}\|t\|_ed|\mu(t)|<\infty\).

MSC:

41A30 Approximation by other special function classes
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
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